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examples of filters
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(Example)
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- If $X$ is any set and $A\subseteq X$ then $\mathcal{F} = \{ F\subseteq X\colon A\subseteq F\}$ is a fixed filter on $X$ ; $\mathcal{F}$ is an ultrafilter iff $A$ consists of a single point.
- If $X$ is any infinite set, then $\{ F\subseteq X\colon X\smallsetminus F \mbox{is finite }\}$ is a free filter on $X$ , called the cofinite filter.
- The filter on $\mathbb{R}$ generated by the filter base $\{ (n,\infty)\colon n\in\mathbb{N}\}$ is called the Fréchet filter on $\mathbb{R}$ ; it is a free filter which does not converge or have any accumulation points.
- The filter on $\mathbb{R}$ generated by the filter base $\{ (0,\varepsilon )\colon\varepsilon >0\}$ is a free filter on $\mathbb{R}$ which converges to $0$ .
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"examples of filters" is owned by Evandar.
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cofinite filter, Fréchet filter |
This object's parent.
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Cross-references: accumulation points, converge, filter base, generated by, filter, free filter, infinite set, point, iff, ultrafilter, fixed filter
There are 3 references to this entry.
This is version 5 of examples of filters, born on 2002-08-02, modified 2004-03-31.
Object id is 3259, canonical name is ExampleOfFilters.
Accessed 5730 times total.
Classification:
| AMS MSC: | 54A99 (General topology :: Generalities :: Miscellaneous) | | | 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous) |
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Pending Errata and Addenda
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